A Generalized Poincare Inequality for a Class of Constant Coefficient Differential Operators

TL;DR

This paper investigates conditions under which a generalized Poincaré inequality holds for first order constant coefficient differential operators, establishing the constant rank condition as sufficient using the Moore-Penrose inverse.

Contribution

It introduces a sufficient condition (constant rank) for the generalized Poincaré inequality to hold for a class of differential operators with constant coefficients.

Findings

Constant rank condition is sufficient for the inequality.

Uses Moore-Penrose inverse in the analysis.

Provides a generalized framework for Poincaré inequalities.

Abstract

We study first order differential operators with constant coefficients. The main question is under what conditions a generalized Poincar\'e inequality holds. We show that the constant rank condition is sufficient. The concept of the Moore-Penrose generalized inverse of a matrix comes into play.